Abstract
Given a set $X$, we provide the algebraic counterpart of the (mixed) Reidemeister moves for virtual knots and links, with semi-arcs labeled by $X$: we define (commutative and noncommutative) invariants with values in groups, using ``2-cocycles, and we also introduce a universal group $U_{nc}^{fg}(X)$ and functions $\pi_f, \pi_g\colon X\times X\to U_{nc}^{fg}(X)$ governing all 2-cocycles in $X$. We exhibit examples of computations -of the group and their invariants- achieved using GAP [7].
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